# Where does the graph of y=(5x^4)-(x^5) have an inflection point?

May 5, 2015

An inflection point is a point on the graph at which the concavity changes. To investigate concavity, we'll look at the sign of the second derivative:

$y = 5 {x}^{4} - {x}^{5}$

$y ' = 20 {x}^{3} - 5 {x}^{4}$

$y ' ' = 60 {x}^{2} - 20 {x}^{3} = 20 {x}^{2} \left(3 - x\right)$

Obviously $20 {x}^{2}$ is always positive, so the sign of $y ' '$ is the same as the sign of $3 - x$.
Which is positive for $x < 3$ and negative for $x > 3$. At $x = 3$ the concavity changes.

An inflection point is a point on the graph, so we need:

when $x = 3$, we get
$y = 5 \left({3}^{4}\right) - {3}^{5} = 5 \left({3}^{4}\right) - 3 \left({3}^{4}\right) = 2 \left({3}^{4}\right) = 2 \left(81\right) = 162$

The point $\left(3 , 162\right)$ in the only inflection point for the graph of $y = 5 {x}^{4} - {x}^{5}$